Question

finding composite area

the window shown is the shape of a semicircle with a radius of 6 feet. the distance from f to e is 3 feet and the measure of $widehat{bc} = 45^circ$. find the area of the glass in region bcih, rounded to the nearest square foot. $square$ ft²

Explanation:

Answer:

To find the area of region \( BCIH \), we recognize it as a sector of a ring (annulus) with central angle \( 45^\circ \).

1. Outer radius (\( R \)): The radius of the larger semicircle is \( 6 \) feet, so \( R = 6 \) ft.
2. Inner radius (\( r \)): The distance from \( F \) to \( E \) is \( 3 \) feet, so the inner radius \( r = 6 - 3 = 3 \) ft (since \( GE = 6 \) ft and \( FE = 3 \) ft, the inner semicircle has radius \( 3 \) ft).
3. Central angle (\( \theta \)): \( \theta = 45^\circ = \frac{\pi}{4} \) radians (or we can use the fraction \( \frac{45}{360} = \frac{1}{8} \) of a full circle).

The area of a sector of a ring (annular sector) is given by:
\[
\text{Area} = \frac{\theta}{2\pi} \times \pi (R^2 - r^2) = \frac{\theta}{2} (R^2 - r^2)
\]

Substituting \( \theta = 45^\circ = \frac{\pi}{4} \) (or \( \frac{1}{8} \) of a full circle), \( R = 6 \), and \( r = 3 \):

\[
\text{Area} = \frac{45^\circ}{360^\circ} \times \pi (6^2 - 3^2) = \frac{1}{8} \times \pi (36 - 9) = \frac{1}{8} \times \pi \times 27
\]

Calculating this:
\[
\text{Area} = \frac{27\pi}{8} \approx \frac{27 \times 3.1416}{8} \approx \frac{84.8232}{8} \approx 10.6029
\]

Rounded to the nearest square foot, the area is \( 11 \) square feet.

\boxed{11}